 reserve j for set;
 reserve p,r for Real;
 reserve S,T,F for RealNormSpace;
 reserve x0 for Point of S;
 reserve g for PartFunc of S,T;
 reserve c for constant sequence of S;
 reserve R for RestFunc of S,T;
 reserve G for RealNormSpace-Sequence;
 reserve i for Element of dom G;
 reserve f for PartFunc of product G,F;
 reserve x for Element of product G;
reserve G for RealNormSpace-Sequence;
reserve F for RealNormSpace;
reserve i for Element of dom G;
reserve f,f1,f2 for PartFunc of product G, F;
reserve x for Point of product G;
reserve X for set;

theorem Th23:
for G be RealNormSpace-Sequence,
    i be Element of dom G,
    x be Point of product G,
    Z be Subset of product G
      st Z is open & x in Z holds
ex N be Neighbourhood of proj(i).x st
  for z be Point of G.i st z in N holds (reproj(i,x)).z in Z
proof
   let G be RealNormSpace-Sequence,
       i be Element of dom G,
       x be Point of product G,
       Z be Subset of product G;
   assume Z is open & x in Z; then
   consider r be Real such that
A1: 0 < r & {y where y is Point of product G : ||. y - x .|| < r} c= Z
      by NDIFF_1:3;
   set N = {y where y is Point of G.i : ||. y - proj(i).x .|| < r};
   reconsider N as Neighbourhood of proj(i).x by A1,NFCONT_1:3;
   take N;
   thus for z be Point of G.i st z in N holds (reproj(i,x)).z in Z
   proof
    let z be Point of G.i;
    assume z in N; then
A2: ex y be Point of G.i st y = z & ||. y - proj(i).x .|| < r;
    ||. (reproj(i,x)).z - x .||
      = ||. reproj(i,0.(product G)).(z - proj(i).x) .|| by Th22
     .= ||. z - proj(i).x .|| by Th21; then
    (reproj(i,x)).z
     in {y where y is Point of product G : ||. y - x .|| < r} by A2;
    hence thesis by A1;
   end;
end;
